Here we explore the properties of the band-spin texture in momentum space for a ferromagnetic (FM) vdW bilayer of CrI\(_3\) , based on first-principles calculations.
This system exemplifies the effects of chirality and electric polarization combined with FM order in establishing the spin-texture features of a relativistic centrosymmetric magnet.
CrI\(_3\)  is a particularly interesting example of 2D vdW magnetic semiconductors, with perpendicular magnetization easy axis. The CrI\(_{3}\)  monolayer is FM, whereas the CrI\(_{3}\)  bilayer in its most common experimental form is a layered antiferromagnet,[1]}, [2]} with a weak interlayer exchange coupling.
This makes possible switching experimentally from the AFM to the FM phase with low magnetic field,[3]} and most remarkably also
by electrical means, such as electrostatic doping and electric field.[2]}, [5]} Change in the layer stacking was also shown to change the magnetic coupling from AFM to FM.[6]}, [7]}, [8]}, [9]}
CrI\(_3\)  possesses rather strong Cr magnetic moments oriented perpendicular to the layers and a hefty Iodine SOC.[10]} Earlier studies of the bilayer and monolayer have shown that SOC has a major impact on the highest valence states, while the lowest conduction bands remain essentially unaffected by relativistic effects.[10]}, [12]}
